Scattering Equations and Global Duality of Residues

Abstract

We examine the polynomial form of the scattering equations by means of computational algebraic geometry. The scattering equations are the backbone of the Cachazo-He-Yuan (CHY) representation of the S-matrix. We explain how the Bezoutian matrix facilitates the calculation of amplitudes in the CHY formalism, without explicitly solving the scattering equations or summing over the individual residues. Since for $n$-particle scattering, the size of the Bezoutian matrix grows only as $(n-3)\times (n-3)$, our algorithm is very efficient for analytic and numeric amplitude computations.

21 pages, v2: journal version

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Cited by in corpus (37)

• Aspects of Scattering Amplitudes and Moduli Space Localization
• On the computation of intersection numbers for twisted cocycles
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Elliptic scattering equations
• Elimination and recursions in the scattering equations
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• Analytic Representations of Yang-Mills Amplitudes
• General Solution of the Scattering Equations
• Comments on the evaluation of massless scattering
• CHY Loop Integrands from Holomorphic Forms
• The Polynomial Form of the Scattering Equations is an H-Basis
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• MultivariateResidues: a Mathematica package for computing multivariate residues
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Feynman Rules of Higher-order Poles in CHY Construction
• Algebraic geometry and Bethe ansatz (I) the quotient ring for BAE
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• A Combinatoric Shortcut to Evaluate CHY-forms
• CHY formula and MHV amplitudes
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• Bootstrapping solutions of scattering equations
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Analytical amplitudes from numerical solutions of the scattering equations
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• Note on the Labelled tree graphs
• Ambitwistor strings and reggeon amplitudes in N=4 SYM
• Note on solutions of scattering equations
• Analytic expressions of amplitudes by the cross-ratio identity method
• MultivariateResidues - a Mathematica package for computing multivariate residues
• CHY Theory for Several Fields
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• Relations and representations of QCD amplitudes